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Properties

Label 1638.4.a
Level $1638$
Weight $4$
Character orbit 1638.a
Rep. character $\chi_{1638}(1,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $37$
Sturm bound $1344$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 37 \)
Sturm bound: \(1344\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1638))\).

Total New Old
Modular forms 1024 90 934
Cusp forms 992 90 902
Eisenstein series 32 0 32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(7\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(6\)
\(+\)\(-\)\(+\)\(-\)$+$\(7\)
\(+\)\(-\)\(-\)\(+\)$+$\(7\)
\(+\)\(-\)\(-\)\(-\)$-$\(6\)
\(-\)\(+\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(+\)\(-\)$+$\(5\)
\(-\)\(+\)\(-\)\(+\)$+$\(5\)
\(-\)\(+\)\(-\)\(-\)$-$\(4\)
\(-\)\(-\)\(+\)\(+\)$+$\(8\)
\(-\)\(-\)\(+\)\(-\)$-$\(6\)
\(-\)\(-\)\(-\)\(+\)$-$\(6\)
\(-\)\(-\)\(-\)\(-\)$+$\(8\)
Plus space\(+\)\(48\)
Minus space\(-\)\(42\)

Trace form

\( 90 q + 4 q^{2} + 360 q^{4} - 32 q^{5} + 16 q^{8} + O(q^{10}) \) \( 90 q + 4 q^{2} + 360 q^{4} - 32 q^{5} + 16 q^{8} - 56 q^{11} + 1440 q^{16} - 204 q^{17} - 260 q^{19} - 128 q^{20} + 72 q^{22} - 536 q^{23} + 1682 q^{25} + 144 q^{29} + 336 q^{31} + 64 q^{32} + 360 q^{34} + 56 q^{35} + 468 q^{37} - 696 q^{38} - 500 q^{41} + 468 q^{43} - 224 q^{44} - 880 q^{46} - 1048 q^{47} + 4410 q^{49} + 156 q^{50} + 1168 q^{53} + 3008 q^{55} + 632 q^{58} - 244 q^{59} + 1072 q^{61} - 32 q^{62} + 5760 q^{64} + 936 q^{65} + 5112 q^{67} - 816 q^{68} - 1288 q^{70} + 2208 q^{71} + 2636 q^{73} + 64 q^{74} - 1040 q^{76} + 3048 q^{79} - 512 q^{80} + 3320 q^{82} + 108 q^{83} - 3224 q^{85} - 512 q^{86} + 288 q^{88} - 1724 q^{89} - 182 q^{91} - 2144 q^{92} + 1152 q^{94} + 3868 q^{95} + 2164 q^{97} + 196 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1638))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7 13
1638.4.a.a 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(-16\) \(7\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-2^{4}q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.b 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(-9\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-9q^{5}-7q^{7}-8q^{8}+\cdots\)
1638.4.a.c 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(-3\) \(-7\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-3q^{5}-7q^{7}-8q^{8}+\cdots\)
1638.4.a.d 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(-3\) \(7\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-3q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.e 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(5\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
1638.4.a.f 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(9\) \(7\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+9q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.g 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(12\) \(7\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+12q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.h 1638.a 1.a $1$ $96.645$ \(\Q\) None \(-2\) \(0\) \(14\) \(-7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+14q^{5}-7q^{7}-8q^{8}+\cdots\)
1638.4.a.i 1638.a 1.a $1$ $96.645$ \(\Q\) None \(2\) \(0\) \(-12\) \(7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-12q^{5}+7q^{7}+8q^{8}+\cdots\)
1638.4.a.j 1638.a 1.a $1$ $96.645$ \(\Q\) None \(2\) \(0\) \(0\) \(7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+7q^{7}+8q^{8}-39q^{11}+\cdots\)
1638.4.a.k 1638.a 1.a $1$ $96.645$ \(\Q\) None \(2\) \(0\) \(3\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+3q^{5}-7q^{7}+8q^{8}+\cdots\)
1638.4.a.l 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{1401}) \) None \(-4\) \(0\) \(-5\) \(-14\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2-\beta )q^{5}-7q^{7}+\cdots\)
1638.4.a.m 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{105}) \) None \(-4\) \(0\) \(-5\) \(-14\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2-\beta )q^{5}-7q^{7}+\cdots\)
1638.4.a.n 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{43}) \) None \(-4\) \(0\) \(-4\) \(14\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2+\beta )q^{5}+7q^{7}+\cdots\)
1638.4.a.o 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{2}) \) None \(4\) \(0\) \(-6\) \(-14\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-3+\beta )q^{5}-7q^{7}+\cdots\)
1638.4.a.p 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{1169}) \) None \(4\) \(0\) \(-5\) \(14\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-2-\beta )q^{5}+7q^{7}+\cdots\)
1638.4.a.q 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{673}) \) None \(4\) \(0\) \(-3\) \(-14\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-1-\beta )q^{5}-7q^{7}+\cdots\)
1638.4.a.r 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{129}) \) None \(4\) \(0\) \(-1\) \(-14\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-\beta q^{5}-7q^{7}+8q^{8}+\cdots\)
1638.4.a.s 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{21}) \) None \(4\) \(0\) \(8\) \(-14\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4+\beta )q^{5}-7q^{7}+8q^{8}+\cdots\)
1638.4.a.t 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{3}) \) None \(4\) \(0\) \(8\) \(14\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+7q^{7}+8q^{8}+\cdots\)
1638.4.a.u 1638.a 1.a $2$ $96.645$ \(\Q(\sqrt{65}) \) None \(4\) \(0\) \(15\) \(14\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(7-\beta )q^{5}+7q^{7}+8q^{8}+\cdots\)
1638.4.a.v 1638.a 1.a $3$ $96.645$ 3.3.118088.1 None \(-6\) \(0\) \(-7\) \(-21\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
1638.4.a.w 1638.a 1.a $3$ $96.645$ 3.3.58908.1 None \(-6\) \(0\) \(-6\) \(-21\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2-\beta _{1})q^{5}-7q^{7}+\cdots\)
1638.4.a.x 1638.a 1.a $3$ $96.645$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-6\) \(0\) \(1\) \(21\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+\beta _{1}q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.y 1638.a 1.a $3$ $96.645$ 3.3.842136.1 None \(-6\) \(0\) \(2\) \(-21\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(1-\beta _{1})q^{5}-7q^{7}+\cdots\)
1638.4.a.z 1638.a 1.a $3$ $96.645$ 3.3.294825.1 None \(6\) \(0\) \(-13\) \(-21\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-5+\beta _{1}-\beta _{2})q^{5}+\cdots\)
1638.4.a.ba 1638.a 1.a $3$ $96.645$ 3.3.360321.1 None \(6\) \(0\) \(-13\) \(21\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-4+\beta _{2})q^{5}+7q^{7}+\cdots\)
1638.4.a.bb 1638.a 1.a $3$ $96.645$ 3.3.1600113.1 None \(6\) \(0\) \(0\) \(-21\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-\beta _{1}q^{5}-7q^{7}+8q^{8}+\cdots\)
1638.4.a.bc 1638.a 1.a $3$ $96.645$ 3.3.58908.1 None \(6\) \(0\) \(6\) \(-21\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(2+\beta _{1})q^{5}-7q^{7}+\cdots\)
1638.4.a.bd 1638.a 1.a $3$ $96.645$ 3.3.7441.1 None \(6\) \(0\) \(6\) \(21\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(2-\beta _{1})q^{5}+7q^{7}+\cdots\)
1638.4.a.be 1638.a 1.a $4$ $96.645$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-10\) \(28\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-3-\beta _{1})q^{5}+7q^{7}+\cdots\)
1638.4.a.bf 1638.a 1.a $4$ $96.645$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(29\) \(28\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(7+\beta _{1})q^{5}+7q^{7}+\cdots\)
1638.4.a.bg 1638.a 1.a $4$ $96.645$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(0\) \(-29\) \(28\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-7-\beta _{1})q^{5}+7q^{7}+\cdots\)
1638.4.a.bh 1638.a 1.a $5$ $96.645$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(0\) \(-19\) \(-35\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-4+\beta _{1})q^{5}-7q^{7}+\cdots\)
1638.4.a.bi 1638.a 1.a $5$ $96.645$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(0\) \(-1\) \(35\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-\beta _{1}q^{5}+7q^{7}-8q^{8}+\cdots\)
1638.4.a.bj 1638.a 1.a $5$ $96.645$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(0\) \(1\) \(35\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+\beta _{1}q^{5}+7q^{7}+8q^{8}+\cdots\)
1638.4.a.bk 1638.a 1.a $5$ $96.645$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(0\) \(19\) \(-35\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4-\beta _{1})q^{5}-7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1638))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1638)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(546))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(819))\)\(^{\oplus 2}\)